Determination of Relative Tensor Orientations by γ-encoded Chemical Shift Anisotropy/Heteronuclear Dipolar Coupling 3D NMR Spectroscopy in Biological Solids

Abstract

In this paper, we present 3D chemical shift anisotropy (CSA)/dipolar coupling correlation experiments, based on γ-encoded R-type symmetry sequences. The γ-encoded correlation spectra are exquisitely sensitive to the relative orientation of the CSA and dipolar tensors and can provide important structural and dynamic information in peptides and proteins. We show that the first-order (m = ±1) and second-order (m = ±2) Hamiltonians in the R-symmetry recoupling sequences give rise to different correlation patterns due to their different dependencies on the crystallite orientation. The relative orientation between CSA and dipolar tensors can be determined by fitting the corresponding correlation patterns. The orientation of ¹⁵N CSA tensor in the quasi-molecular frame is determined by the relative Euler angles, α_NH and β_NH, when the combined symmetry schemes are applied for orientational studies of ¹H-¹⁵N dipolar and ¹⁵N CSA tensors. The correlation experiments introduced here work at moderate magic angle spinning frequencies (10-20 kHz) and allow for simultaneous measurement of multiple sites of interest. We studied the orientational sensitivity of γ-encoded symmetry-based recoupling techniques numerically and experimentally. The results are demonstrated on [¹⁵N]-N-acetyl-valine (NAV) and N-formyl-Met-Leu-Phe (MLF) tripeptide.

Introduction

Three-dimensional structures of peptides, proteins and protein assemblies at atomic-level resolution provide important clues for understanding their biological function. Solid-state nuclear magnetic resonance (SSNMR) is emerging as an important technique for structural elucidation of biological systems,¹ and is particularly powerful for studies of large biopolymers, such as membrane proteins,^2-8 amyloid fibrils,^9-12 intact viruses,^{13, 14} and protein assemblies.^{15, 16} Even for globular, soluble proteins, SSNMR is becoming an essential structural tool since solid-state NMR observables inform about properties of the system at hand that are unavailable from solution measurements.^17-26 For example, the anisotropic spin interactions, such as chemical shift (CS) and dipolar interactions, bear a wealth of information about the geometric and electronic structure of a wide variety of solid systems including organic polymers and proteins.^{18, 25-34} In MAS experiments, these anisotropic spin interactions are averaged out partially or completely by the rapid spinning of the sample around the fixed axis at “magic angle”. In order to measure and exploit these anisotropic interactions, SSNMR recoupling methods have been developed that restore the geometric and electronic structure information.^35-49 Alternatively, anisotropic interactions can also be measured under static conditions; however, site resolution is limited in the solid powders, and for multiple sites in a system, it is advantageous to carry out such measurements on aligned samples.^50-54

Among the numerous recoupling techniques,^{35-44, 55} one class of methods, developed by M. Levitt,^56-59 employs a symmetry-based principle for the design of pulse sequences that are based on the use of rotor-synchronized radio-frequency (rf) fields in conjunction with magic angle spinning. There are two main symmetry-based pulse sequence classes, CN_n^v and RN_n^v, where the symbols N, n and v are small integers and are referred to as the symmetry numbers of the pulse sequence. The CN_n^v sequences are designed such, that no net rotation of the spin angular momenta is produced as the result of the rotor-synchronized train of radiofrequency pulses, the so-called the C blocks. In the RN_n^v sequences, a net rotation of the spin angular momenta by π around the x axis of the rotating frame is created. Comparing these two classes of symmetry-based sequences, the R-type sequences have more restrictive selection rules than the C-type sequences.^{40, 59, 60} The choice of any particular symmetry-based recoupling sequence depends on the tensorial interaction of interest and on the experimental conditions (i.e., the range of MAS frequencies and rf fields) available to the NMR spectroscopist.

Knowledge about the magnitude and orientation of the ¹⁵N chemical shift anisotropy (CSA) tensor is important for biological solids because it provides critical information on the primary/secondary structure, electrostatics, hydrogen bonding, solvation, and dynamics.^{54, 61, 62} Although the magnitudes of the three principal elements of the ¹⁵N CSA tensor can be measured easily with existing methods in non-single-crystalline solids under MAS conditions, these experiments do not provide any direct information about the orientation of the CSA tensor in the molecular frame. In contrast, for single crystalline solids, both the magnitude and orientation of the three principal elements of the CSA tensor can be determined by the NMR measurement.⁶² However, many interesting biological systems (e.g., membrane proteins, protein assemblies, and other biopolymers) cannot be prepared as single crystals, and therefore development of methods for determination of the orientation of the CSA tensor in such noncrystalline samples under MAS conditions (where the resolution and the sensitivity of the measurements is the highest) is becoming more and more important.

For orienting the ¹⁵N CSA tensor in the molecular frame, the three Euler angles (α, β, γ) are required, but there is no SSNMR approach for determining γ directly. However, since to an excellent approximation, the δ_zz principal component of the ¹⁵N CSA tensor lies in the peptide plane and the Euler angle γ is usually close to zero, it is reasonable to postulate that the orientation of ¹⁵N CSA tensor is determined by α and β in the quasi-molecular frame. Based on two- and three-dimensional experiments for correlating chemical shift and dipolar tensors, several methods have been proposed for characterization of the CSA tensor orientation in the quasi-molecular frame, and these methods have been applied successfully to a number of powder samples.^{50, 51, 63-68} PISEMA (Polarization Inversion Spin Exchange at the Magic Angle)⁶⁶ and COAST-C7₂¹ (Combined offset and anisotropy stabilization)⁶³ can be used to extract the orientation of the CSA tensor, however these types of experiments are performed under static or slow magic angle spinning conditions, making them suitable for studying only a single site of interest. For a system containing multiple sites, PISEMA or COAST-C7₂¹ suffer from peak overlap, interfering with the simultaneous determination of the relative orientations of ¹⁵N CSA tensors in a site-resolved manner.

In this work, we demonstrate a three-dimensional MAS NMR approach based on γ-encoded R-type symmetry sequences, in which dipolar and CSA interactions are re-introduced into the t₁ and t₂ indirect dimensions, respectively. The resulting correlation spectra are especially sensitive to the relative orientation of the CSA and dipolar tensors. The experiments introduced here work at moderate MAS frequencies (10-20 kHz) and yield orientational information simultaneously for multiple sites. The orientational sensitivity of γ-encoded symmetry-based recoupling techniques was studied numerically and experimentally. Results are demonstrated on [¹⁵N]-N-acetyl-valine (NAV) and N-formyl-Met-Leu-Phe (MLF) tripeptide.

Theoretical background: a γ-encoded symmetry-based recoupling approach

Consider a system of coupled S spins undergoing a mechanical rotation about a fixed axis. The spin Hamiltonian of such system in the presence of an rf pulse irradiation consists of the time-dependent rf Hamiltonian and the time-dependent internal Hamiltonian, and can be expressed as^{58, 60}

$$H(t)=H_{\mathit{\text{rf}}}(t)+H_{int}(t)$$ (1)

$$H_{int}(t)=\sum _{\Lambda ,l,m,\lambda ,\mu }{H}_{\mathit{\text{lm}}\lambda \mu }^{\Lambda }(t)$$ (2)

with the quantum numbers l and λ indicating the rank of the spin interaction Λ with respect to spatial rotation and spin rotations, respectively. m (-l, -l+1, …, +l) and μ (-λ - λ+1, …, +λ) refer to the space and spin rotational component, respectively. The rotation of spins caused by the rf irradiation results in the internal Hamiltonian elements with μ ≠ 0. With the Magnus expansion of the effective Hamiltonian in the interaction frame, the first-order average internal Hamiltonian for t = 0∼τ is given by

$$\begin{array}{ll}{\overline{H}}^{\Lambda }\hfill & =\sum _{l,m,\lambda ,\mu }{\overline{H}}_{\mathit{\text{lm}}\lambda \mu }^{\Lambda }\hfill \\ \hfill & =\sum _{l,m,\lambda ,\mu }{K}_{m\lambda \mu }{A}_{\mathit{\text{lm}}}^{\Lambda }({\Omega }_{\mathit{\text{PR}}})d_{m0}^l({\beta }_{\mathit{\text{RL}}})T_{\lambda \mu }^{\Lambda }\hfill \end{array}$$ (3)

The scaling factor due to the rf irradiation is expressed as

$$K_{m\lambda \mu }={\tau }^{-1}{\int }_0^{\tau }\mathit{\text{dt}}\cdot e^{\mathit{\text{im}}{\varpi }_{r}t+i\mu {\gamma }_{\mathit{\text{rf}}}(t)}{d}_{\mu 0}^{\lambda }[-{\beta }_{\mathit{\text{rf}}}(t)]$$ (4)

With the irradiation by the periodic RN_n^v symmetry pulses, the selection rules for the first-order average Hamiltonian satisfy the following equation:

$${\overline{H}}_{\mathit{\text{lm}}\lambda \mu }^{\Lambda }=0\text{if}\mathit{\text{mn}}-\mu v\ne \frac{N}{2}{Z}_{\lambda }$$ (4)

with Z_λ an integer with the same parity as λ. On the basis of the different rotational properties of the spin interactions, a given type of interaction can be recoupled selectively by designing the appropriate symmetry pulse sequences. Considering a symmetry-allowed spin Hamiltonian characterized by a particular set of quantum numbers {l, m, λ, μ}, the component $A_{\mathit{\text{lm}}}^{\Lambda }({\Omega }_{\mathit{\text{PR}}})$ of the spatial tensor in equation (3) has the following form:

$$A_{\mathit{\text{lm}}}^{\Lambda }({\Omega }_{\mathit{\text{PR}}})={\sum _{m″m\prime }[A_{\mathit{\text{lm}}}^{\Lambda }]}^P{D}_{m″m\prime }^l({\Omega }_{\mathit{\text{PM}}}^{\Lambda })D_{m\prime m}^l({\Omega }_{\mathit{\text{MR}}})$$ (6)

$$D_{m\prime m}^l({\Omega }_{\mathit{\text{MR}}})=exp\{-\mathit{\text{im}}\prime {\alpha }_{\mathit{\text{MR}}}\}{d}_{m\prime m}^l({\beta }_{\mathit{\text{MR}}})exp\{-\mathit{\text{im}}{\gamma }_{\mathit{\text{MR}}}\}$$ (7)

The magnitude of each single component of the recoupled Hamiltonian is independent of the Euler angle γ_MR, and it is only the phase that is γ_MR-dependent. γ-encoding generally leads to a high recoupling efficiency in a powder and good discrimination of the recoupled spin interactions.^{40, 69, 70} In addition, γ-encoded recoupling methods are highly sensitive to the molecular orientation, since only the recoupled lineshape and the magnitude depend on two Euler angles, α_MR and β_MR, which in turn describe the molecular orientation in the rotor frame. Although the complete recoupled spin Hamiltonian usually contains several symmetry-allowed terms which result in a non- γ-encoded recoupled Hamitonian, it is possible to design the selection rules for each spatial quantum number associated with a single spin quantum number, thereby maintaining the entire recoupled Hamiltonian γ-encoded.

For example, the R18₁⁷ symmetry sequence suppresses the isotropic chemical shift, heteronuclear J-coupling and homonuclear dipolar interactions, but recouples the CSA and heteronuclear dipole-dipole interactions characterized by quantum numbers {l, m, λ, μ} = {2, 2, 1,-1} and {2, -2, 1, 1}.⁴⁰ Since each spatial component is associated with only a single spin component, the spin Hamitonian recoupled by R18₁⁷ is γ-encoded. Figure 1 represents the selection properties for CSA and heteronuclear dipolar interactions in two other symmetry-based recoupling sequences, R14₂³ and R14₂⁵. As shown in Figure 1(a), the only Hamiltonian terms that are symmetry-allowed by R14₂³ are {l, m, λ, μ} = {2, 2, 1, -1} and {2, -2, 1, 1}. It therefore is a γ-encoded recoupling sequence with the same selection rules as R18₁⁷. The R14₂⁵ sequence exhibits different selection properties for the spin interactions with l = 2 and λ = 1. In R14₂⁵, the symmetry-allowed terms are {l, m, λ, μ} = {2, 1, 1, -1} and {2, -1, 1, 1}, and this sequence is also γ-encoded for recoupling of the CSA or heteronuclear dipolar interactions. At the same time, the spin Hamiltonian terms with quantum numbers {l, m, λ, μ} = {2, 2, 2, -2} and {2, -2, 2, 2} are also symmetry-allowed in R14₂⁵, indicating that it too is γ-encoded for recoupling the homonuclear dipolar interaction. R14₂⁵ therefore is a powerful sequence for the determination of heteronuclear dipolar or CSA interactions in systems with weak homonuclear dipolar interactions. In addition, R14₂⁵ provides higher scaling factors than R14₂³ and R18₁⁷.

Figure 1.

Space-spin selection diagrams for R14₂³ (a) and R14₂⁵ (b). (a) R14₂³ selects the components of the heteronuclear dipolar and CSA interactions with quantum numbers {l, m, λ, μ} = {2, 2, 1, -1} and {2, -2, 1, 1}, respectively. (b) R14₂⁵ selects the components of the heteronuclear dipolar and CSA interactions with quantum numbers {l, m, λ, μ} = {2, 1, 1, -1} and {2, -1, 1, 1}, respectively.

It is important to note that for the recoupling of the CSA interactions, the space-spin selections dictate which terms in the Hamiltonian will be recovered, and these selection rules are different for each specific symmetry-based sequence. The average CSA Hamiltonian in Equation (3) can be separated into three components, corresponding to the zeroth-order (m = 0), the first-order (m = ±1) and the second-order (m = ±2) terms according to the spatial rank number, respectively:⁶⁹

$${\overline{H}}_{(0)}^{\mathit{\text{CSA}}}=\sum _{m=0,\lambda ,\mu }{K}_{m\lambda \mu }{A}_{\mathit{\text{lm}}}^{\Lambda }({\Omega }_{\mathit{\text{PR}}})d_{m0}^l({\beta }_{\mathit{\text{RL}}})T_{\lambda \mu }^{\Lambda }=\gamma B_0{\sigma }_0{I}_z$$ (8)

$${\overline{H}}_{(1)}^{\mathit{\text{CSA}}}=\sum _{m=\pm 1,\lambda ,\mu }{K}_{m\lambda \mu }{A}_{\mathit{\text{lm}}}^{\Lambda }({\Omega }_{\mathit{\text{PR}}})d_{m0}^l({\beta }_{\mathit{\text{RL}}})T_{\lambda \mu }^{\Lambda }=\gamma B_0{\sigma }_{1}cos({\gamma }_{\mathit{\text{PR}}}+{\omega }_{r}t+{\delta }_1)I_z$$ (9)

$${\overline{H}}_{(2)}^{\mathit{\text{CSA}}}=\sum _{m=\pm 2,\lambda ,\mu }{K}_{m\lambda \mu }{A}_{\mathit{\text{lm}}}^{\Lambda }({\Omega }_{\mathit{\text{PR}}})d_{m0}^l({\beta }_{\mathit{\text{RL}}})T_{\lambda \mu }^{\Lambda }=\gamma B_0{\sigma }_{2}cos2({\gamma }_{\mathit{\text{PR}}}+{\omega }_{r}t+{\delta }_2)I_z$$ (10)

Here the expressions for σ₀, σ₁, and σ₂ depend on the crystallite orientation and the three principal values of the CSA tensor. It can be seen that the first-order and the second-order CSA Hamiltonian terms can be removed by sample spinning at the magic angle, and a sharp line is observed at σ₀. However, rotor-synchronized pulse irradiation of correct symmetry would selectively recover the CSA Hamiltonian terms. According to the quantum numbers of the symmetry-allowed Hamiltonian terms, R14₂⁵ and R14₂³ can be referred to as the first-order σ₁ and the second-order σ₂ γ-encoded symmetry sequences, respectively. In this nomenclature, σ₀, σ₁, and σ₂ are expressed as

$$\begin{array}{l}{\sigma }_0={\sigma }^{\mathit{\text{iso}}}+P_2(cos{\beta }_{\mathit{\text{RL}}})[P_2(cos{\beta }_{\mathit{\text{PR}}})({\sigma }_{\mathit{\text{zz}}}^{\mathit{\text{PAS}}}-{\sigma }^{\mathit{\text{iso}}})+\frac{1}{2}sin{\beta }_{\mathit{\text{PR}}}cos2{\alpha }_{\mathit{\text{PR}}}({\sigma }_{\mathit{\text{xx}}}^{\mathit{\text{PAS}}}-{\sigma }_{\mathit{\text{yy}}}^{\mathit{\text{PAS}}})]\end{array}$$ (11)

$$\begin{array}{ll}{\sigma }_1\hfill & =\sqrt{{\sigma }_{\mathit{\text{COS}}1}^2+{\sigma }_{\mathit{\text{SIN}}1}^2}\hfill \\ \hfill & =sin{\beta }_{\mathit{\text{RL}}}cos{\beta }_{\mathit{\text{RL}}}sin{\beta }_{\mathit{\text{PR}}}{[{\{cos{\beta }_{\mathit{\text{PR}}}[-3({\sigma }_{\mathit{\text{zz}}}^{\mathit{\text{PAS}}}-{\sigma }^{\mathit{\text{iso}}})+cos2{\alpha }_{\mathit{\text{PR}}}({\sigma }_{\mathit{\text{xx}}}^{\mathit{\text{PAS}}}-{\sigma }_{\mathit{\text{yy}}}^{\mathit{\text{PAS}}})]\}}^2+{[sin2{\alpha }_{\mathit{\text{PR}}}({\sigma }_{\mathit{\text{xx}}}^{\mathit{\text{PAS}}}-{\sigma }_{\mathit{\text{yy}}}^{\mathit{\text{PAS}}})]}^2]}^{1/2}\hfill \end{array}$$ (12)

$$\begin{array}{ll}{\sigma }_2\hfill & =\sqrt{{\sigma }_{\mathit{\text{COS}}2}^2+{\sigma }_{\mathit{\text{SIN}}2}^2}\hfill \\ \hfill & =½{sin}^2{\beta }_{\mathit{\text{RL}}}{\{{[\frac{3}{2}{sin}^2{\beta }_{\mathit{\text{PR}}}({\sigma }_{\mathit{\text{zz}}}^{\mathit{\text{PAS}}}-{\sigma }^{\mathit{\text{iso}}})+\frac{1}{2}(1+cos2{\beta }_{PR})\times cos2{\alpha }_{\mathit{\text{PR}}}({\sigma }_{\mathit{\text{xx}}}^{\mathit{\text{PAS}}}-{\sigma }_{\mathit{\text{yy}}}^{\mathit{\text{PAS}}})]}^2+{[cos{\beta }_{\mathit{\text{PR}}}sin2{\alpha }_{\mathit{\text{PR}}}({\sigma }_{\mathit{\text{xx}}}^{\mathit{\text{PAS}}}-{\sigma }_{\mathit{\text{yy}}}^{\mathit{\text{PAS}}})]}^2\}}^{1/2}\hfill \end{array}$$ (13)

$$\begin{array}{ll}{\delta }_1\hfill & ={tan}^{-1}(\frac{{\sigma }_{\mathit{\text{SIN}}1}}{{\sigma }_{\mathit{\text{COS}}1}})\hfill \\ \hfill & ={tan}^{-1}\left[\frac{sin2{\alpha }_{\mathit{\text{PR}}}({\sigma }_{\mathit{\text{xx}}}^{\mathit{\text{PAS}}}-{\sigma }_{\mathit{\text{yy}}}^{\mathit{\text{PAS}}})}{cos{\beta }_{\mathit{\text{PR}}}[-3({\sigma }_{\mathit{\text{zz}}}^{\mathit{\text{PAS}}}-{\sigma }^{\mathit{\text{iso}}})+cos2{\alpha }_{\mathit{\text{PR}}}({\sigma }_{\mathit{\text{xx}}}^{\mathit{\text{PAS}}}-{\sigma }_{\mathit{\text{yy}}}^{\mathit{\text{PAS}}})]}\right]\hfill \end{array}$$ (14)

$$\begin{array}{ll}{\delta }_2\hfill & =½{tan}^{-1}(\frac{{\sigma }_{\mathit{\text{SIN}}2}}{{\sigma }_{\mathit{\text{COS}}2}})\hfill \\ \hfill & =½{tan}^{-1}\left[\frac{cos{\beta }_{\mathit{\text{PR}}}sin2{\alpha }_{\mathit{\text{PR}}}({\sigma }_{\mathit{\text{xx}}}^{\mathit{\text{PAS}}}-{\sigma }_{\mathit{\text{yy}}}^{\mathit{\text{PAS}}})}{\frac{3}{2}{sin}^2{\beta }_{\mathit{\text{PR}}}({\sigma }_{\mathit{\text{zz}}}^{\mathit{\text{PAS}}}-{\sigma }^{\mathit{\text{iso}}})+\frac{1}{2}(1+cos2{\beta }_{\mathit{\text{PR}}})\times cos2{\alpha }_{\mathit{\text{PR}}}({\sigma }_{\mathit{\text{xx}}}^{\mathit{\text{PAS}}}-{\sigma }_{\mathit{\text{yy}}}^{\mathit{\text{PAS}}})}\right]\hfill \end{array}$$ (15)

Where the Euler angles (α_PR, β_PR, γ_PR) define the rotational transformation from the pincipal axis system (PAS) to the rotor fixed frame (RFF), and the principal element σ_ii ^PAS (i = x, y, z) denotes the element of the chemical shift tensor in the PAS. σ_COSi and σ_SINi (i = 1 and 2) are the projected elements of the i^th-order CSA Hamiltonian (σ_i) in x′y′ plane of the RFF, δ_i denotes the projection angle of CSA Hamiltonian σ_i relative to the x′ axis of the RFF.

Experimental section

Materials

The ¹⁵N labeled N-acetyl-valine (NAV) and ¹³C,¹⁵N-labeled N-formyl-Met-Leu-Phe (MLF) tripeptide were purchased from Cambridge Isotope Laboratories. Both powder samples were packed into the MAS rotors for subsequent NMR experiments without any further purification or recrystallization.

NMR Experiments

All NMR experiments were carried out on a TECMAG Discovery spectrometer operating at a resonance frequency of 400.49 MHz for ¹H, 100.64 MHz for ¹³C, and 40.55 MHz for ¹⁵N. A 3.2 mm Varian triple-resonance probehead was used. The 3D pulse sequence that was used for correlating the dipolar and CSA tensors is displayed in Figure 2. All spectra were recorded at room temperature with a MAS frequency of 10 kHz. The typical 90° pulse lengths were 3.3 μs for ¹H and 4.8 μs for ¹⁵N. The cross-polarization contact time was 1.0 ms, and the recycle delay was 3.0 s. Continuous wave (CW) ¹H decoupling with an rf field strength of 76 kHz was performed during the acquisition period and the residual part of the rotational echo period. During the R14₂^3/5 recoupling period, ¹H TPPM decoupling with an rf field strength of 76 kHz and ¹³C XiX decoupling with an rf field strength of 69 kHz were carried out simultaneously. RF field strengths were 90 and 35 kHz for the R18₁⁷ and R14₂^3/5 irradiation periods, respectively. The indirect dimensions (t₁ and t₂) of the 3D MAS experiments were incremented rotor-synchronously, by 100 μs and 200 μs, respectively. A total of 16 t₁ and 32 t₂ increments were used to record the 3D spectra, and 32 and 64 scans were accumulated for each increment, for N-acetyl-valine and N-formyl-Met-Leu-Phe, respectively. The data were processed by zero-filling and Fourier transformation in NMRPipe in a Mac environment. Zero-filling up to 256 points was performed in t₁ and t₂ dimensions, and line broadening of 100 Hz was applied in ω₁ and ω₂ during the processing.

Figure 2.

Pulse sequence for the three-dimensional CSA/DIP orientational correlation experiment. Solid and blank squares denote π/2 and π pulses, respectively. ¹H-¹⁵N dipolar coupling and ¹⁵N CSA interactions are reintroduced by the γ-encoded symmetry sequences R18₁⁷ and R14₂^5/3 in the indirect t₁ and t₂ dimensions, respectively.

Simulations of the NMR spectra

Numerical simulations of 1D and 2D NMR spectra were performed using the SIMPSON software package.⁷¹ 986 ZCW angles (α, β) and 3 γ angles were used to generate a powder average. The ¹H Larmor frequency was set as 400.5 MHz, and a MAS frequency of 10 kHz was used for all simulations. The applied RF field intensities were 90 and 35 kHz for R18₁⁷ and R14₂^3/5 irradiation, respectively. NMR parameters that are required as an input for the program are H-N dipolar couplings and the components of the ¹H and ¹⁵N chemical shift anisotropy tensor, δ_σ and η. ¹H CSA with δ_σ = 8.9 ppm and η = 0.75 was used for all simulations. Since the C-N heteronuclear dipolar couplings are completely averaged out by the XiX decoupling, carbon spins were not considered in any of the simulations. In order to determine the accurate relative orientation between the tensors, identical acquisition and processing parameters were used as in the simulations and the NMR experiments.

For defining the chemical shift tensor we follow the Haeberlen-Mehring-Spiess convention, where the isotropic value δ_iso, the reduced anisotropy δ_σ, the anisotropy Ω, and the asymmetry parameter η, are related to the principal elements of the chemical shift tensor according to |δ_zz −δ_iso|≥|δ_xx −δ_iso|≥|δ_yy −δ_iso|, δ_iso =⅓(δ_xx + δ_yy +δ_zz), δ_σ = δ_zz −δ_iso = ⅔Ω., and η = (δ_yy−δ_xx)/(δ_zz −δ_iso).

Density Functional Theory Calculations

¹⁵N magnetic shielding anisotropy tensor calculations were performed in Gaussian03⁷² using the B3LYP functional⁷³ and tzvp basis set.⁷⁴ The atomic coordinates were taken from the solid-state NMR structure of MLF⁷⁵ and used in the single-point calculations of the magnetic shielding anisotropy tensors without further geometry optimization. The calculated principal components of the magnetic shielding tensor, σ_ii (i = 1, 2, or 3), were converted to the principal components of the chemical shift tensors, δ_ii, using the relationship δ_ii = σ_iso(ref) - σ_ii, where σ_iso(ref) = 244.6 ppm is the absolute shielding for liquid ammonia at 25 °C.⁷⁶

Results and Discussion

Simulated DIP/CSA Correlation Spectra

Figure 3 illustrates the orientation of the principal axes of the amide nitrogen CSA in the common molecular frame. Euler angles α_NH and β_NH determine the relative orientation between the ¹⁵N CSA and the ¹H-¹⁵N dipolar tensors; this orientation yields important structural and dynamic information (i.e., torsion angle ϕ and backbone internal motional timescales in proteins). The relative orientation can be determined by correlating the H-N dipolar coupling, along the first indirect dimension ω₁, and the chemical shift anisotropy of the amide nitrogen, along the second indirect dimension ω₂. The 3D pulse sequence shown in Figure 2 was used to record such a 2D correlation spectrum extracted along the ¹⁵N isotropic chemical shift dimension. After the ¹H-¹⁵N cross polarization, the ¹⁵N magnetization evolves during the t₁ period under the R18₁⁷ Hamiltonian applied in the ¹H channel; this Hamiltonian reintroduces the ¹H-¹⁵N dipolar couplings and the ¹H CSA interactions while suppressing the ¹H-¹H homonuclear dipolar couplings. Subsequently, the dipolar-modulated ¹⁵N magnetization is tilted into the longitudinal direction and is subjected to the rf pulse irradiation of the R14₂³ or R14₂⁵ symmetry.

Figure 3.

(a) Molecular structure of a peptide unit depicting the orientation of chemical shift tensor of the amide nitrogen in the molecular frame, defined by the N-H bond (z-axis) and the C-N-H peptide plane. (b) Euler rotations between the principal axis system (PAS) of the CSA tensor and the molecular frame. Euler angles α_NH and β_NH represent the orientation of the three principal elements of ¹⁵N CSA tensor in the peptide unit. The peptide plane contains the N-H bond and the δ_zz principal component of the ¹⁵N CSA tensor as shown in (b), with a zero Euler angle γ.

As discussed above, the R14₂³ sequence has the same space-spin selection pathways and the same scaling factors as R18₁⁷, where the ¹H-¹⁵N/¹³C-¹⁵N heteronuclear dipolar couplings and the ¹⁵N anisotropic chemical shift interactions are reintroduced. In order to achieve pure ¹⁵N CSA interactions during the R14₂³ pulse train period, TPPM and XiX heteronuclear decoupling is applied to protons and carbons, respectively. Usually, for efficient ¹H or ¹³C decoupling a decoupling rf field strength greater than twice the rf field strength of the R14₂^3/5 element (3.5×ω_R) is required. The R14₂⁵ symmetry sequence also recouples the ¹⁵N-¹⁵N homonuclear dipolar couplings while reintroducing the heteronuclear dipole-dipole and CSA interactions. It should be noted, however, that any effects of the ¹⁵N-¹⁵N dipolar couplings are negligible, given their small sizes (tens of Hertz). Therefore, the dipolar-modulated ¹⁵N magnetization nutates under σ₁ or σ₂ γ-encoded CSA Hamiltonian during t₂, and is detected during t₃ in the presence of ¹H proton CW decoupling. The resulting 3D spectrum correlates ¹H-¹⁵N dipolar couplings with the ¹⁵N CSA interactions in two dimensions, and, in the third dimension, the isotropic ¹⁵N chemical shift. Therefore it is easy to extract a 2D spectrum that correlates the dipolar and the CSA interactions along the ¹⁵N isotropic chemical shift.

We analyzed the R-symmetry-based DIP/CSA correlation spectra for determining the relative orientation of the two tensors, using the SIMPSON simulation package. The simulated 2D DIP/CSA correlation spectra are shown in Figure 4 and correspond to different combined R-symmetry sequences. For both simulations, R18₁⁷ is used for recoupling the ¹H-¹⁵N dipolar interactions, while R14₂⁵ and R14₂³ are employed for recoupling the ¹⁵N σ₁ and σ₂ CSA interactions as depicted in Figure 4a and 4b, respectively. To simulate the spectra, we constructed a molecular model consisting of two coupled (¹H, ¹⁵N) spins with the same internal interaction parameters. For each spin, ¹H-¹⁵N dipolar coupling and ¹H (or ¹⁵N) chemical shift anisotropy tensors were included in the simulation. For the spectra shown in Figure 4, the principal frame of the ¹⁵N chemical shift tensor was coincident with that of the anisotropic ¹H-¹⁵N dipole-dipole interaction, i.e., α_NH = 0° and β_NH = 0°. Due to the different orientational dependence in the σ₂-CSA recoupled R14₂³ and σ₁-CSA recoupled R14₂⁵, different correlation spectra result from these two recoupling sequences as demonstrated in Figure 4 (a) and (b), respectively, even though the same R18 sequence is used for the ¹H-¹⁵N dipolar recoupling during t₁ dimension.

Figure 4.

Simulated CSA/DIP correlation NMR spectra produced by combined symmetry-based sequences, R18₁⁷/R14₂³ (a) and R18₁⁷/R14₂⁵ (b). The CSA and dipolar tensors are assumed to be coincident in this simulation (α_NH = 0° and β_NH = 0°). 64 t₁ increments and 128 t₂ increments were used to generate the spectra.

Since the splitting of the recoupled lineshapes by γ-encoded R18₁⁷ and R14₂^3/5 sequences depends greatly on the crystallite orientation in the laboratory frame (while the non γ-encoded recoupling schemes do not exhibit such orientational dependence), the 2D DIP/CSA correlation spectrum is sensitive to the relative orientation between the two tensors both in the crystalline and in the powder samples (simulated data are shown in the supporting information).

Figure 5 displays the simulated 2D R18₁⁷/R14₂³ DIP/CSA correlation spectra as the function of the relative tensor orientations; the corresponding R18₁⁷/R14₂⁵ spectra are shown in the Supporting Inforamation. From the series of these simulated spectra, it is clear that the powder lineshapes representing the DIP/CSA correlation are extremely sensitive to the orientation of the most deshielded element δ_zz relative to the N-H bond, represented by the Euler angle β_NH. The Euler angle α_NH affects the 2D correlation spectrum as well, but its effect on the lineshape is much less pronounced (simulated data shown in the Supporting Information). In all simulations, the δ_xx principal value of the CS A tensor is considered to lie in the Cα-N-H peptide plane (α_NH = 0°). With the increase of the Euler angle β_NH, the lineshape spreads over a larger frequency range. These simulated 2D correlation plots demonstrate that it should be possible to extract the relative orientation between the dipolar and CSA tensors from the corresponding experimental spectra.

Figure 5.

Simulated R18₁⁷/R14₂³ CSA/DIP spectra for different relative tensor orientations. In these simulations, the δ_xx principal value CSA is considered to lie in the Cα-N-H peptide plane (α_NH = 0°), and the angle β_NH between the H-N vector and the CSA δ_zz principal value was changed from 0 to 70 degrees with 10-degree increments from (a) to (h), respectively. 64 t₁ increments and 128 t₂ increments were used for each simulation. A powder average was generated with 986 ZCW angles and 3 γ angles.

Experimental DIP/CSA Correlation Spectra

The applications of R18₁⁷ for determining heteronuclear dipolar couplings have been reported previously.^{26, 32, 40} However, the measurement of σ₁ or σ₂ CSA by R14₂^3/5 has not yet been investigated. Figure 6 shows the simulated and the experimental ¹⁵N σ₁ and σ₂ CSA spectra of NAV generated by R14₂^3/5 symmetry sequences using an RF field intensity of 35 kHz for R14 irradiation. It can be seen that the scaling factor for recoupling the σ₁ CSA by R14₂⁵ is higher than that for recoupling the σ₂ CSA by R14₂³, although the σ₁ and σ₂ CSA lineshapes are similar. The central peak in the experimental spectra is assigned to the effect of finite ¹⁵N relaxation times. The simulations yield a best-fit reduced anisotropy δ_σ of 103 ppm with an asymmetry parameter η of 0.25, and the simulated lineshapes agree well with the experimental spectra. These CSA parameters are also consistent with values reported previously.^{62, 77} Our results demonstrate that γ-encoded R14₂^3/5 experiments can be performed to successfully determine the ¹⁵N CSA tensors, and that R14₂⁵ yields good accuracy in systems with weak homonuclear dipolar interactions (vide supra).

Figure 6.

Experimental (solid lines) and simulated (dashed lines) σ₁ and σ₂ CSA recoupled ¹⁵N NMR spectra of [¹⁵N]-N-acetyl-valine (NAV) generated by γ-encoded R14₂⁵ (a) and R14₂³(b) sequences, respectively. The sample was spun at an MAS frequency of 10 kHz. In both 2D experiments, 32 scans were accumulated for each t₁ increment. A total of 32 t₁ increments were used. The acquisition and processing parameters used in SIMPSON simulations were the same as those utilized in the corresponding NMR experiments. The best-fit simulated spectra were obtained with δ_σ = 103 ppm and η = 0.25.

The experimental ¹H-¹⁵N DIP/¹⁵N CSA 2D correlation spectra of NAV obtained with the combined R18₁⁷/R14₂^3/5 symmetry sequences are shown in Figures 7 (a) and 8 (a), respectively. The 2D lineshapes were extracted from the 3D correlation experiment (vide supra) along the ¹⁵N isotropic chemical shift in the ω₃ dimension. R14₂⁵ and R14₂³ were performed to reintroduce the ¹⁵N σ₁- (Figure 7(a)) and σ₂- (Figure 8(a)) CSA powder patterns, respectively. Consistent with the simulations discussed above, the experimental correlation spectra are different due to the different dependence of the lineshapes in each recoupling sequence on the relative tensor orientations. There also exists a central peak in the 2D correlation spectra arising from the ¹⁵N relaxation, and its contribution to the overall spectrum makes it difficult to extract the DIP/CSA cross peaks cleanly, as shown in Figure 8(a). In order to determine the relative orientation between the ¹⁵N CSA and ¹H-¹⁵N dipolar tensors, a series of simulations are performed with Euler angles (α and β) monotonously increasing by 5°. Figures 7(b) and 8(b) show the best-fit simulated DIP/CSA correlation spectra for the 3D R18₁⁷/R14₂⁵ and R18₁⁷/R14₂³ sequences, respectively. The parameters used in the simulations, ¹H-¹⁵N dipolar coupling (ω_D(H-N) = 10.4 kHz) and the ¹⁵N CSA (δ_σ = 103 ppm and η = 0.25) were extracted from the simulated 1D R18₁⁷ dipolar and R14₂⁵ CSA patterns shown in Figure 6. For the R18₁⁷/R14₂⁵ experiment, the best-fit relative orientation was obtained for β_NH(δ_zz) = 25°±5° and α_NH(δ_xx) = 30°±5°. The best-fit simulated pattern was chosen visually, and the final error is estimated to be in range of ±5°. These angles establish unambiguously the orientation of ¹⁵N CSA tensor in the quasi-molecular frame. Similarly, the best-fit relative orientation (β_NH(δ_zz) = 25°±5° and α_NH(δ_xx) = 20°±10°) was obtained for the 3D R18₁⁷/R14₂³ experiment. The consistency of the relative tensor orientations extracted from the two independent experiments is encouraging and illustrates the accuracy of the current approach. Figure 7(c, d) and Figure 8(c, d) show the DIP/CSA correlation patterns for other relative orientations. The orientation of ¹⁵N CSA tensor in the molecular frame determined by the present NMR method agrees well with the previous reports.^{62, 77} As can be observed, the ridge patterns depend to a large degree on the orientation of δ_zz relative to the N-H bond, and it is clear that there is a unique orientation that is consistent with the NMR experiment. Here it should be noted that the librational and vibrational motions would give rise to reduced CSA and dipolar interactions,⁷⁸ and this is commonly exploited in the studies of dynamics by CSA and dipolar recoupling techniques. These motions also add to the uncertainty in the experiments for measuring the relative orientation between CSA and dipolar tensors. A motionally averaged relative orientation would result in this case, especially for the system with librational motions where the relative orientation varies continuously over time.

Figure 7.

(a) 2D ¹⁵N CSA/¹H-¹⁵N DIP correlation spectrum of N-acetyl-valine (NAV), extracted from the 3D R18₁⁷/R14₂⁵ experiment along the ¹⁵N isotropic chemical shift. The spectrum was recorded with 64 scans, 16 t₁ and 32 t₂ increments. The sample was spun at a MAS frequency of 10 kHz. (b) Best-fit simulated correlation spectrum, obtained with β_NH(δ_zz) = 25°±5° and α_NH(δ_xx) = 30°±10°, using the same acquisition and processing parameters as in the NMR experiment in (a). (c) Simulated correlation spectrum with β_NH(δ_zz) = 0° and α_NH (δ_xx) = 0°. (d) Simulated correlation spectrum with β_NH(δ_zz) = 90° and α_NH(δ_xx) = 0°.

Figure 8.

(a) 2D ¹⁵N CSA/¹H-¹⁵N DIP correlation spectrum of N-acetyl-valine (NAV), extracted from the 3D R18₁⁷/R14₂³ experiment along the ¹⁵N isotropic chemical shift. The spectrum was recorded with 64 scans, 16 t₁ and 32 t₂ increments. The sample was spun at a MAS frequency of 10 kHz. (b) Best-fit simulated correlation spectrum, obtained with β_NH(δ_zz) = 25°±5° and α_NH(δ_xx) = 20°±5°, and the same acquisition and processing parameters as in the NMR experiment. (c) Simulated correlation spectrum with β_NH(δ_zz) = 0° and α_NH(δ_xx) = 0°. (d) Simulated correlation spectrum with β_NH(δ_zz) = 90° and α_NH(δ_xx) = 0°.

The present combined symmetry sequences can also be applicable for simultaneous determination of DIP-CSA relative tensor orientations for multiple sites of interest. Figure 9 shows the ¹⁵N σ₁ and σ₂ CSA recoupled NMR powder patterns of ¹³C,¹⁵N-enriched MLF tripeptide generated by γ-encoded R14₂³ and R14₂⁵ symmetry sequences; these powder patterns were extracted along the ¹⁵N isotropic chemical shifts of each residue. The ¹⁵N CSA parameters for each site can be derived from the experimental spectra using SIMPSON simulations. In Table 1, the best-fit ¹⁵N CSA experimental parameters for MLF generated by the R14₂^3/5 symmetry-based recoupling sequences are compared with those obtained from quantum chemical calculations using DFT. It can be seen that the experimental ¹⁵N CSA parameters derived from the R14₂^3/5 symmetry sequences generally agree with DFT computations very well. We note that there are small differences between the ¹⁵N σ₁ and σ₂ CSA parameters extracted from the two pulse sequences, which we attribute to the effect of the ¹⁵N relaxation on the central peak, as discussed above.

Figure 9.

¹⁵N σ₁ and σ₂ CSA recoupled NMR spectra of ¹³C, ¹⁵N N-formyl-Met-Leu-Phe (MLF) tripeptide, for Phe (a, d), Leu (b, e), and Met (c, f) residues. The spectra were recorded using R14₂⁵ (a - c) and R14₂³ (d - e) pulse sequences, respectively. The sample was spun at a MAS frequency of 10 kHz. The experimental spectra are shown in black, the best-fit spectra simulated in SIMPSON, in red. The best-fit chemical shift anisotropy δ_σ and asymmetry parameter η as well as the comparison with other experimental and computational results are presented in Table 1.

Table 1.

¹⁵N CSA parameters for each residue in ¹³C, ¹⁵N N-formyl-MLF: experimental derived from R14₂^3/5 recoupling sequences and calculated using Density Functional Theory. The experimental and simulated σ₁ and σ₂ CSA powder patterns are shown in Figure 9.

Residue	Method	δiso(ppm)	δσppm)	ησ
Met	R1425	125.5	107±1	0.33±0.01
	R1423	125.5	109±1	0.32±0.01
	DFT: b3lyp/tzvp	117.3	110.4	0.40
Leu	R1425	116.2	93±1	0.34±0.01
	R1423	116.2	97±1	0.35±0.01
	DFT: b3lyp/tzvp	112.6	93.2	0.70
Phe	R1425	107.6	101±1	0.32±0.01
	R1423	107.6	101±1	0.34±0.01
	DFT: b3lyp/tzvp	103.7	97.0	0.40

Three-dimensional R18₁⁷/R14₂^3/5 experiments were also carried out on the MLF tripeptide for measuring the ¹⁵N CSA orientation for each residue, as shown in Figure 10 (a – f). As discussed above, R18₁⁷ and R14₂³ or R14₂⁵ are used for correlating the ¹H-¹⁵N dipolar coupling and the ¹⁵N σ₂ or σ₁ CSA interactions, respectively. 2D DIP/CSA correlation spectra for each residue were extracted from these 3D data sets along the ¹⁵N isotropic chemical shift dimension (ω₃). SIMPSON simulations were carried out to derive the relative ¹H-¹⁵N dipolar and ¹⁵N CSA tensor orientations using the best-fit calculated ¹H-¹⁵N dipolar coupling and the ¹⁵N CSA parameters for each site, as shown in Table 2. The simulated best-fit DIP/CSA correlation powder patterns for each residue are shown in Figure 10 (g – l). Table 2 also lists the parameters obtained experimentally and computed using DFT. The comparison of experimental and DFT-calculated ¹⁵N CSA parameters shown in Table 1, reveals that these are in good agreement, giving us confidence in both the experimental and computational methodologies. Although the ¹⁵N orientation information obtained by DFT using b3lyp/tzvp method is slightly different from that derived from the symmetry-based NMR measurement, the difference is within the experimental error (see Table 2). Our results demonstrate that the amide nitrogen of each residue in the MLF tripeptide has a similar CSA tensor orientation relative to the H-N-Cα plane. The most deshielded element of the amide ¹⁵N CSA tensor, δ_zz, is close to, but non-coincident with the H-N bond.

Figure 10.

Experimental (a-f) and simulated (g-l) ¹⁵N CSA/¹H-¹⁵N dipolar correlation spectra of ¹³C, ¹⁵N N-formyl-Met-Leu-Phe (MLF) tripeptide, extracted from the corresponding 3D experiments along the ¹⁵N isotropic chemical shift for Phe (a, d), Leu (b, e) and Met (c, f) residues. 3D R18₁⁷/R14₂⁵ and R18₁⁷/R14₂³ pulse sequences were used to record the experimental (a - c) and (d - f) spectra, respectively, and the same sequences were employed to generate the simulated (g – i) and (j – l) spectra, respectively. The sample was spun at a MAS frequency of 10 kHz. ¹H-¹⁵N dipolar and ¹⁵N CSA parameters used in the simulations were extracted from 1D R18 dipolar and 1D R14 CSA pattern simulations, and fixed during the calculations. The orientation of the ¹⁵N CSA relative to ¹H-¹⁵N dipolar tensor was optimized. The best-fit relative orientations between the ¹H-¹⁵N dipolar and ¹⁵N CSA are shown in Table 2.

Table 2.

The relative orientations between ¹H-¹⁵N dipolar and ¹⁵N CSA tensors for each residue in ¹³C, ¹⁵N N-formyl-MLF: experimental derived from R18₁⁷/R14₂^3/5 recoupling sequences and calculated using Density Functional Theory.

Residue	Method	αNH (degrees)	βNH (degrees)
Met	R1817/R1423/5	20±10	20±5
	DFT: b3lyp/tzvp	9.8	14.8
Leu	R1817/R1423/5	20±10	25±5
	DFT: b3lyp/tzvp	6.3	14.7
Phe	R1817/R1423/5	10±10	35±10
	DFT: b3lyp/tzvp	24	19

The experiments discussed above are generally applicable to peptides and proteins. The time for the 3D experiments introduced in this work is similar to that required for any 3D experiments on uniformly enriched proteins, and therefore, the same sensitivity considerations apply. The protein samples have to be stable over extended periods of time (days and up to several weeks), and instrument stability (rf fields, temperature, MAS frequency) is highly important. With the rapid progress in hardware (probes), data collection and sample preparation protocols, such as fast-MAS technology,^79-81 non-uniform sampling,^{82, 83} use of paramagnetically doped samples to reduce the ¹H spin-lattice relaxation times and corresponding recycle delays without affecting other NMR observables,⁸⁴ as well as proton detection^{6, 85} dramatic sensitivity enhancements are observed in multidimensional protein spectroscopy. A combination of these approaches enables 3D spectra to be detected in several hours rather than several days as we have seen in our laboratory (data to be reported elsewhere), making it possible to study very large and/or dilute proteins and protein assemblies. The information about the relative orientations of the CSA and dipolar tensors derived from the measurements introduced in this work is expected to be invaluable for accurate measurement of torsion angles, which in turn represent an important constraint in tertiary structure determination of peptides and proteins in the solid state.^{86, 87}

Conclusions

In summary, combined R-symmetry sequences can be used to measure the relative orientations between the ¹H-¹⁵N dipolar and ¹⁵N chemical shift anisotropy tensors. γ-encoded R14₂⁵ and R14₂³ symmetry sequences are used to reintroduce the first-order (σ₁) and the second-order (σ₂) CSA powder patterns, respectively. These sequences work best at moderate MAS frequencies, which is the regime most commonly used for structural and dynamics studies in uniformly and extensively isotopically enriched peptides, proteins and protein assemblies. Each of these symmetry-based recoupling blocks can be combined with the R18₁⁷ dipolar recoupling sequence to construct 2D dipolar/CSA correlation patterns. The ridges of these correlation spectra are very sensitive to the relative orientation between the two tensors. Since the dipolar and CSA principal components can be readily determined by independent measurements, the analysis of the spectral DIP/CSA correlation patterns is straightforward, with Euler angles β_NH(δ_zz) and α_NH(δ_xx) being the only variable parameters. The latter can be readily extracted from the numerical simulations of the spectra. Our results show that the β_NH(δ_zz) angle has the highest sensitivity to correlation patterns. The central peaks due to the ¹⁵N relaxation contribute to the 1D/2D spectral lineshapes; these relaxation effects are necessary to be taken into account in the spectral simulations. Work is in progress on incorporating the effect of relaxation into the spectral simulations and on designing experiments that would minimize its influence on the experimental lineshapes. By incorporating a ¹⁵N isotropic chemical shift dimension into a 3D experiment, these sequences become applicable for simultaneous determination of multiple sites in peptides and proteins, and are expected to become a useful tool for measuring torsion angles, important constraints in protein structure determination in the solid state.